Clubs were told at a meeting that no postponements would be permitted if 13 outfield players and one goalkeeper from their 25-man squad list were available, according to the Daily Mail.
Premier League clubs have decided against a temporary halt to the season, despite a surge in coronavirus cases that has forced mass postponements.
Just four of the weekend's scheduled 10 games went ahead as Britain battles record numbers of positive Covid cases due to the new Omicron strain.
The Premier League board has been considering applications for postponements on a case-by-case basis but there has been criticism over the decision-making process, with fears it jeopardises the sporting integrity of the competition.
According to the Daily Mail, clubs were told at a meeting on Monday that no postponements would be permitted if 13 outfield players and one goalkeeper from their 25-man squad list were available.
Chelsea, who have fallen six points behind leaders Manchester City at the top of the table, asked to have Sunday's match at Wolves called off but their plea was rejected despite seven positive cases.
Title-chasing Liverpool also stumbled in a 2-2 draw at Tottenham, with a number of key players absent with coronavirus.
The crisis has come at the busiest time in the English football calendar, with Premier League sides each due to play three rounds of matches between December 26 and January 3.
Even a short circuit-breaker would put strain on the calendar, forcing clubs to cram in fixtures during the second half of the season.
But it appears a proposal to postpone the middle round of the festive fixtures from December 28-30 to give squads stretched by infections and injuries extra preparation time was dismissed at Monday's meeting.
Rick Parry, chairman of the English Football League, which runs the three divisions below the Premier League, has said there is no "scientific evidence" that a firebreaker would work.
Scrapping FA Cup replays and the planned two-legged semi-final of the League Cup in January have been suggested as solutions to create space in the calendar.
A call for more transparency
Arsenal manager Mikel Arteta admitted his side were "lucky" to be in action this weekend as they thumped Leeds 4-1 and repeated his call for more transparency on the decision-making process.
The Gunners were forced to play at Brentford on the opening weekend of the season despite four players testing positive and lost the game 2-0.
Adding to the frustration over the perceived unfairness of call-offs is that some of the clubs with the best vaccination rates have been forced to play on.
Unvaccinated players still have to self-isolate for 10 days after coming into close contact with a positive case, which has further depleted a number of squads.